Trigonometric Graphs

To sketch the sine graph between $0^\circ$ and $360^\circ$, start at the origin $(0, 0)$. The curve rises to a local maximum turning point at $(90^\circ, 1)$, crosses the $x$-axis at $180^\circ$, drops to a local minimum at $(270^\circ, -1)$, and finishes the cycle by crossing the $x$-axis at $360^\circ$.

[Diagram: A sketch of $y = \sin x$ for $0^\circ \le x \le 360^\circ$, showing a smooth continuous wave. The $x$-axis intercepts are labelled at $0^\circ$, $180^\circ$, and $360^\circ$. A maximum turning point is labelled at $(90^\circ, 1)$ and a minimum turning point at $(270^\circ, -1)$.]

The cosine graph is exactly the same shape, but it is translated $90^\circ$ to the left. It starts at its maximum turning point $(0, 1)$, crosses the $x$-axis at $90^\circ$, hits a minimum at $(180^\circ, -1)$, crosses the $x$-axis again at $270^\circ$, and returns to a maximum at $(360^\circ, 1)$.

[Diagram: A sketch of $y = \cos x$ for $0^\circ \le x \le 360^\circ$, showing a smooth continuous wave. The $y$-axis intercept is at $(0, 1)$. The $x$-axis intercepts are labelled at $90^\circ$ and $270^\circ$. A minimum turning point is labelled at $(180^\circ, -1)$ and a maximum at $(360^\circ, 1)$.]

The Tangent Graph

You can easily walk over a gently rolling hill, but what happens when you reach a sheer cliff face that drops away into nothing? The tangent graph ($y = \tan x$) behaves just like this, breaking the continuous wave pattern seen in sine and cosine.

Unlike sine and cosine, the tangent graph is not a continuous wave and has no amplitude, meaning it has no maximum or minimum turning points. Instead, it consists of repeating branches that always have a positive, increasing gradient from $-\infty$ to $+\infty$.

The period of the tangent graph is only $180^\circ$, intercepting the $x$-axis at $0^\circ$, $180^\circ$, and $360^\circ$.

Crucially, the tangent graph features vertical asymptotes at $90^\circ$ and $270^\circ$. An asymptote is a vertical dashed line that the graph approaches but never touches or crosses (entering $\tan(90^\circ)$ on a calculator results in a "Math Error").

[Diagram: A sketch of $y = \tan x$ for $0^\circ \le x \le 360^\circ$, showing three repeating curve branches that pass through $0^\circ$, $180^\circ$, and $360^\circ$ on the $x$-axis. Vertical dashed lines represent asymptotes at $x = 90^\circ$ and $x = 270^\circ$.]

Interpreting and Solving with Trigonometric Graphs

How can a single equation like $\sin x = 0.5$ have more than one correct answer? Because trigonometric graphs are periodic and symmetrical, an equation typically has multiple solutions between $0^\circ$ and $360^\circ$.

You can find these solutions graphically by drawing a horizontal line across your sketch (e.g., at $y = 0.5$) and reading the $x$-values at every intersection point.

Alternatively, you can use the built-in symmetry of the graphs to find the second solution after using your calculator to find the first:

$$\sin(x) = \sin(180^\circ - x)$$ $$\cos(x) = \cos(360^\circ - x)$$ $$\tan(x) = \tan(x + 180^\circ)$$

Worked Example

Solve the equation $\cos x = 0.5$ for the range $0^\circ \le x \le 360^\circ$.

Step 1: Find the first solution using the inverse cosine function.

• $x = \cos^{-1}(0.5)$
• $x = 60^\circ$

Step 2: Use the symmetry of the cosine graph to find the second solution.

• The cosine graph is symmetrical such that $\cos(x) = \cos(360^\circ - x)$.
• $x = 360^\circ - 60^\circ$
• $x = 300^\circ$

Step 3: State all valid solutions within the required range.

• $x = 60^\circ, 300^\circ$